Question

Consider a collection of envelopes consisting of 3 red envelopes​, 3 blue envelopes​, 1 green envelope​, and 3 yellow envelopes. If three envelopes are selected at​ random, without​ replacement, determine the probability that at least one envelope is a red envelope. The probability that at least one envelope is red is

Answer 1

Answer: Our required probability is $\dfrac{31}{35}$

Step-by-step explanation:

Since we have given that

Number of red envelopes = 3

Number of blue envelopes = 3

Number of green envelopes = 1

We need to select 3 envelopes in such a way that at least one envelope is a red.

So, it becomes,

$\dfrac{^3C_1\times ^4C_2}{^7C_3}+\dfrac{^3C_2\times ^4C_1}{^7C_3}+\dfrac{^3C_3}{7C_3}\\\\=\dfrac{18}{35}+\dfrac{12}{35}+\dfrac{1}{35}\\\\=\dfrac{18+12+1}{35}\\\\=\dfrac{31}{35}$

Hence, our required probability is $\dfrac{31}{35}$